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A multifractal formalism for non-concave and non-increasing spectra: the leaders profile method Esser, Céline ; Kleyntssens, Thomas ; Nicolay, Samuel in Applied & Computational Harmonic Analysis (in press) We present an implementation of a multifractal formalism based on the types of histogram of wavelet leaders. This method yields non-concave spectra and is not limited to their increasing part. We show ... [more ▼] We present an implementation of a multifractal formalism based on the types of histogram of wavelet leaders. This method yields non-concave spectra and is not limited to their increasing part. We show both from the theoretical and from the applied points of view that this approach is more e cient than the wavelet-based multifractal formalisms previously introduced. [less ▲] Detailed reference viewed: 26 (10 ULg)Large deviation spectra based on wavelet leaders Bastin, Françoise ; Esser, Céline ; Jaffard, Stéphane in Revista Matemática Iberoamericana (in press) We introduce a new multifractal formalism, based on distributions of wavelet leaders, which allows to detect non-concave and decreasing multifractal spectra, and we investigate the properties of the ... [more ▼] We introduce a new multifractal formalism, based on distributions of wavelet leaders, which allows to detect non-concave and decreasing multifractal spectra, and we investigate the properties of the associated function spaces. [less ▲] Detailed reference viewed: 68 (10 ULg)Estimation de spectres de singularités par des méthodes de grandes déviations basées sur les ondelettes Esser, Céline Scientific conference (2015, December 02) La régularité d'un signal en un point peut être caractérisée par son exposant de Hölder. Une description synthétique de la répartition des différents exposants de Hölder d'un signal est fournie par son ... [more ▼] La régularité d'un signal en un point peut être caractérisée par son exposant de Hölder. Une description synthétique de la répartition des différents exposants de Hölder d'un signal est fournie par son spectre de singularités. Plusieurs méthodes basées sur la décomposition en ondelettes ont été proposées pour estimer le spectre de singularités d'un signal; elles reposent sur une caractérisation de l'exposant de Hölder par les coefficients d'ondelettes. Dans cet exposé, nous présentons une nouvelle méthode reposant sur des estimations de grandes déviations basées sur les coefficients d'ondelettes et montrons son efficacité sur plusieurs exemples déterministes ou non. [less ▲] Detailed reference viewed: 5 (3 ULg)Estimation de spectres de singularités par des techniques de grandes déviations basées sur les ondelettes Esser, Céline Conference (2015, November 26) La régularité en un point d'une fonction peut être caractérisée par son exposant de Hölder. Si la fonction est très irrégulière, celui-ci peut varier fortement d'un point à l'autre. Dans ce cas, il est ... [more ▼] La régularité en un point d'une fonction peut être caractérisée par son exposant de Hölder. Si la fonction est très irrégulière, celui-ci peut varier fortement d'un point à l'autre. Dans ce cas, il est naturel d'étudier la distribution des exposants de Hölder: cette information est donnée par le spectre de singularités de la fonction. En général, ce spectre est assez difficile à calculer et on utilise des méthodes pour l'estimer. Plusieurs méthodes basées sur la décomposition en ondelettes ont été proposées ; elles reposent sur une caractérisation de l'exposant de Hölder par les coefficients d'ondelettes. Dans cet exposé, nous introduisons une nouvelle technique reposant sur des estimations de grandes déviations basées sur les coefficients d'ondelettes et montrons son efficacité sur plusieurs exemples [less ▲] Detailed reference viewed: 8 (0 ULg)Analyse multifractale de la divergence de séries d'ondelettes Esser, Céline Scientific conference (2015, November 06) Dans cet exposé, nous étudions la divergence ponctuelle de séries d'ondelettes dans un espace de Besov. Nous obtenons une borne supérieure pour la dimension de Hausdorff de l'ensemble des points ayant un ... [more ▼] Dans cet exposé, nous étudions la divergence ponctuelle de séries d'ondelettes dans un espace de Besov. Nous obtenons une borne supérieure pour la dimension de Hausdorff de l'ensemble des points ayant un taux de divergence donné, et nous montrons que cette borne est optimale en utilisant les notions de résidualité et de prévalence. Il s'agit d'un travail en collaboration avec S. Jaffard. [less ▲] Detailed reference viewed: 4 (0 ULg)A propos des fonctions continues qui ne sont dérivables en aucun point Esser, Céline Conference (2015, August 03) En 1872, Karl Weierstrass présenta non seulement une, mais toute une famille de fonctions continues et nulle part dérivables. Après la publication de ce résultat, beaucoup d'autres mathématiciens ... [more ▼] En 1872, Karl Weierstrass présenta non seulement une, mais toute une famille de fonctions continues et nulle part dérivables. Après la publication de ce résultat, beaucoup d'autres mathématiciens apportèrent leur propre contribution en construisant d'autres fonctions continues et nulle part dérivables. Dans cet exposé, nous présenterons les fonctions de Weierstrass et nous montrerons que le théorème de Baire permet d'affirmer que l'ensemble des fonctions nulle part dérivables est dense dans l'ensemble des fonctions continues. Nous étudierons également la régularité ponctuelle des fonctions de Weierstrass en introduisant la notion d'exposant de Hölder. [less ▲] Detailed reference viewed: 40 (7 ULg)Denjoy-Carleman classes and lineability Esser, Céline Conference (2015, June 16) The Denjoy-Carleman classes are spaces of smooth functions which satisfy growth conditions on their derivatives defined through weight sequences. In this talk, given a Denjoy-Carleman class E of Beurling ... [more ▼] The Denjoy-Carleman classes are spaces of smooth functions which satisfy growth conditions on their derivatives defined through weight sequences. In this talk, given a Denjoy-Carleman class E of Beurling type that strictly contains another non-quasianalytic class F of Roumieu type, we handle the question of knowing how large the set of functions in E that are nowhere in the class F is. In particular, we prove the dense-lineability of the set of functions of E which are nowhere in F. Consequences for the Gevrey classes are also given. We extend then these results to the case of classes of ultradifferentiable functions defined using weight functions. [less ▲] Detailed reference viewed: 31 (4 ULg)Dense-lineability in classes of ultradifferentiable functions Esser, Céline Conference (2015, May 27) The Denjoy-Carleman classes are spaces of smooth functions which satisfy growth conditions on their derivatives defined through weight sequences. In this talk, given a Denjoy-Carleman class E of Beurling ... [more ▼] The Denjoy-Carleman classes are spaces of smooth functions which satisfy growth conditions on their derivatives defined through weight sequences. In this talk, given a Denjoy-Carleman class E of Beurling type that strictly contains another non-quasianalytic class F of Roumieu type, we handle the question of knowing how large the set of functions in E that are nowhere in the class $F$ is. In particular, we prove the dense-lineability of the set of functions of $E$ which are nowhere in F. Consequences for the Gevrey classes are also given. We extend then these results to the case of classes of ultradifferentiable functions defined imposing conditions on their Fourier Laplace transform using weight functions. [less ▲] Detailed reference viewed: 16 (3 ULg)A propos des fonctions continues nulle part dérivables Esser, Céline Conference (2015, February 24) En 1872, Karl Weierstrass choqua la communauté mathématique en présentant une famille de fonctions continues nulle part dérivables. Après la publication de ce résultat, beaucoup d'autres mathématiciens ... [more ▼] En 1872, Karl Weierstrass choqua la communauté mathématique en présentant une famille de fonctions continues nulle part dérivables. Après la publication de ce résultat, beaucoup d'autres mathématiciens apportèrent leur propre contribution en construisant d'autres fonctions continues et nulle part dérivables. Dans cet exposé, nous présenterons les fonctions de Weierstrass. De plus, nous montrerons que le th eorème de Baire permet d'affirmer que l'ensemble des fonctions nulle part dérivables est dense dans l'ensemble des fonctions continues. Nous étudierons également la régularité ponctuelle de telles fonctions en utilisant la notion d'exposant de Hölder. [less ▲] Detailed reference viewed: 22 (2 ULg)Topology on new sequence spaces defined with wavelet leaders Bastin, Françoise ; Esser, Céline ; Simons, Laurent in Journal of Mathematical Analysis and Applications (2015), 431(1), 317-341 Using wavelet leaders instead of wavelet coefficients, new sequence spaces of type Sν are defined and endowed with a natural topology. Some classical topological properties are then studied; in particular ... [more ▼] Using wavelet leaders instead of wavelet coefficients, new sequence spaces of type Sν are defined and endowed with a natural topology. Some classical topological properties are then studied; in particular, a generic result about the asymptotic repartition of the wavelet leaders of a sequence in Lν is obtained. Eventually, comparisons and links with Oscillation spaces are also presented as well as with Sν spaces. [less ▲] Detailed reference viewed: 53 (20 ULg)Lacunary wavelet series on Cantor sets Esser, Céline E-print/Working paper (2015) We construct functions with prescribed multifractal spectra which satisfy the leaders profile method. Detailed reference viewed: 10 (0 ULg)Algebrability and nowhere Gevrey differentiability Bastin, Françoise ; ; Esser, Céline et al in Israel Journal of Mathematics (2015), 205 We show that there exist c-generated algebras (and dense in C^infty([0,1])) every nonzero element of which is a nowhere Gevrey diff erentiable function. This leads to results of dense algebrability (and ... [more ▼] We show that there exist c-generated algebras (and dense in C^infty([0,1])) every nonzero element of which is a nowhere Gevrey diff erentiable function. This leads to results of dense algebrability (and, therefore, lineability) of functions enjoying this property. In the process of proving these results we also provide a new construction of nowhere Gevrey di fferentiable functions. [less ▲] Detailed reference viewed: 134 (42 ULg)Regularity of functions: Genericity and multifractal analysis Esser, Céline Doctoral thesis (2014) As surprising as it may seem, there exist functions of C∞(R) which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones may exist. A ... [more ▼] As surprising as it may seem, there exist functions of C∞(R) which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones may exist. A classical technique is to use the Baire category theorem and the notion of residuality. This notion is purely topological and does not give any information about the measure of the set of objects satisfying such a property. In this purpose, the notion of prevalence has been introduced. Moreover, one could also wonder whether large algebraic structures of such objects can be constructed. This question is formalized by the notion of lineability. The first objective of the thesis is to go further into the study of nowhere analytic functions. It is known that the set of nowhere analytic functions is residual and lineable in C∞([0, 1]). We prove that the set of nowhere analytic functions is also prevalent in C∞([0, 1]). Those results of genericity are then generalized using Gevrey classes, which can be seen as intermediate between the space of analytic functions and the space of infinitely differentiable functions. We also study how far such results of genericity could be extended to spaces of ultradifferentiable functions, defined using weight sequences or using weight functions. The second main objective is to study the pointwise regularity of functions via their multifractal spectrum. Computing the multifractal spectrum of a function using directly its definition is an unattainable goal in most of the practical cases, but there exist heuristic methods, called multifractal formalisms, which allow to estimate this spectrum and which give satisfactory results in many situations. The Frisch-Parisi conjecture, classically used and based on Besov spaces, presents two disadvantages: it can only hold for spectra that are concave and it can only yield the increasing part of spectra. Concerning the first problem, the use of Snu spaces allows to deal with non-concave increasing spectra. Concerning the second problem, a generalization of the Frisch-Parisi conjecture obtained by replacing the role played by wavelet coefficients by wavelet leaders allows to recover the decreasing part of concave spectra. Our purpose in this thesis is to combine both approaches and define a new formalism derived from large deviations based on statistics of wavelet leaders. As expected, we show that this method yields non-concave spectra and is not limited to their increasing part. From the theoretical point of view, we prove that this formalism is more efficient than the previous wavelet-based multifractal formalisms. We present the underlying function space and endow it with a topology. [less ▲] Detailed reference viewed: 182 (43 ULg)Genericity and classes of ultradifferentiable functions Esser, Céline Conference (2014, September 26) As surprising as it may seem, there exist infinitely differentiable functions which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones ... [more ▼] As surprising as it may seem, there exist infinitely differentiable functions which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones may exist. A classical technique is to use the Baire category theorem and the notion of residuality. This notion is purely topological and does not give any information about the measure of the set of objects satisfying such a property. In this purpose, the notion of prevalence has been introduced. Moreover, one could also wonder whether large algebraic structures of such objects can be constructed. This question is formalized by the notion of lineability. The first objective of this talk is to go further into the study of nowhere analytic functions. It is known that the set of nowhere analytic functions is residual and lineable in C^infty([0, 1]). We prove that the set of nowhere analytic functions is also prevalent in this space. Those results of genericity are then generalized using Gevrey classes, which can be seen as intermediate between the space of analytic functions and the space of infinitely differentiable functions. We also study how far such results of genericity could be extended to spaces of ultradifferentiable functions, defined using weight sequences. [less ▲] Detailed reference viewed: 10 (0 ULg)Detection of non-concave and non-increasing spectra: Snu spaces revisited with wavelet leaders Esser, Céline Conference (2014, September 26) Our objective is to study the pointwise regularity of functions via their multifractal spectrum. Computing the multifractal spectrum of a function using directly its definition is an unattainable goal in ... [more ▼] Our objective is to study the pointwise regularity of functions via their multifractal spectrum. Computing the multifractal spectrum of a function using directly its definition is an unattainable goal in most of the practical cases, but there exist heuristic methods, called multifractal formalisms, which allow to estimate this spectrum and which give satisfactory results in many situations. The Frisch-Parisi conjecture, classically used and based on Besov spaces, presents two disadvantages: it can only hold for spectra that are concave and it can only yield the increasing part of spectra. Concerning the first problem, the use of S spaces allows to deal with non-concave increasing spectra. Concerning the second problem, a generalization of the Frisch-Parisi conjecture obtained by replacing the role played by wavelet coefficients by wavelet leaders allows to recover the decreasing part of concave spectra. We present a combination of both approaches to define a new formalism derived from large deviations based on statistics of wavelet leaders. We also present the associated function space. [less ▲] Detailed reference viewed: 22 (5 ULg)Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part I) Esser, Céline ; Kleyntssens, Thomas ; Bastin, Françoise et al Conference (2014, May 22) Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to ... [more ▼] Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to determine the spectrum of singularities of the signal, which is the Hausdor ff dimension of the set of points which have the same H ölder exponent. For real-life signals, the computation of the spectrum of singularities from its de finition is not feasible. Multifractal formalisms are used to approximate this spectrum. Currently, there exist several methods. In this talk, we present a new multifractal formalism based on the wavelet leaders of a signal which allows to detect non concave and non increasing spectra. [less ▲] Detailed reference viewed: 44 (7 ULg)Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part II) Kleyntssens, Thomas ; Esser, Céline ; Nicolay, Samuel Conference (2014, May 22) This talk follows "Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part I)" given by Céline Esser. A multifractal formalism is a numerically computable formula that ... [more ▼] This talk follows "Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part I)" given by Céline Esser. A multifractal formalism is a numerically computable formula that approximates the spectrum of singularities of a function. A new multifractal formalism based on the wavelet leaders is presented as well as a comparison with other formalisms. Its main advantages are that it allows to detect non concave and non increasing spectra. An implementation is proposed. [less ▲] Detailed reference viewed: 49 (18 ULg)A new multifractal formalism based on wavelet leaders : detection of non concave and non increasing spectra (Part I) Esser, Céline ; Kleyntssens, Thomas ; Nicolay, Samuel et al Conference (2014, March 25) Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to ... [more ▼] Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to determine the spectrum of singularities of the signal, which is the Hausdorff dimension of the set of points which have the same Hölder exponent. The spectrum of singularities of many mathematical functions can be determined directly from its definition. However, for many real-life signals, the numerical determination of their Hölder regularity is not feasible. Therefore, one cannot expect to have a direct access to their spectrum of singularities and one has to find an indirect way to compute it. A multifractal formalism is a formula which is expected to yield the spectrum of singularities from quantities which are numerically computable. Several multifractal formalisms based on the wavelet coefficients of a signal have been proposed to estimate its spectrum. The most widespread of these formulas is the so-called thermodynamic multifractal formalism, based on the Frish-Parisi conjecture. This formalism presents two drawbacks: it can hold only for spectra that are concave and it can yield only the increasing part of the spectrum. This first problem can be avoided using Snu spaces. The second one can be avoided using a formalism based on wavelet leaders of the signal. In this talk, we propose a new multifractal formalism, based on a generalization of the Snu spaces using wavelet leaders. It allows to detect non concave and non increasing spectra. An implementation of this method is presented in the talk "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II)" of T. Kleyntssens. [less ▲] Detailed reference viewed: 76 (13 ULg)A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II) Kleyntssens, Thomas ; Esser, Céline ; Nicolay, Samuel Conference (2014, March 25) This talk follows "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part I)" given by Céline Esser. For real-life signals, it is impossible to ... [more ▼] This talk follows "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part I)" given by Céline Esser. For real-life signals, it is impossible to compute the spectrum of singularities by using its definition. A multifractal formalism is used to approximate this spectrum. We present a new multifractal formalism for non concave and non increasing spectra based on wavelet leaders. In this talk, an implementation of this formalism is given and several numerical examples are presented. [less ▲] Detailed reference viewed: 49 (16 ULg)Revisiting Snu spaces with wavelet leaders to detect non concave and non increasing spectra Esser, Céline Poster (2014, February 27) A multifractal formalism is a formula which is expected to yield the spectrum of singularities of a function from quantities which are numerically computable. The most widespread of these formulas is the ... [more ▼] A multifractal formalism is a formula which is expected to yield the spectrum of singularities of a function from quantities which are numerically computable. The most widespread of these formulas is the so-called thermodynamic mul- tifractal formalism, based on the Frish-Parisi conjecture. It presents two drawbacks: it can hold only for spectra which are concave and it can only yield the increasing part of the spectrum. This first problem can be avoided using Sν spaces. The second one can be taken care using the wavelet leaders method. In this poster, we present a new multifractal formalism based on a generaliza- tion of the Sν spaces using wavelet leaders. It allows to detect non concave and non increasing spectra. We compare this formalism with the Sν method and the wavelet leaders method. It is based on joint works with F. Bastin, S. Jaffard, T. Kleyntssens and S. Nicolay. [less ▲] Detailed reference viewed: 59 (32 ULg) |
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